非負實係數多項式都能不能寫成平方和一直都被猜測，直到希爾伯特證明出任意n變數非負齊2d次多項式，都可以寫成一些d次多項式的平方和的充分必要條件為(a) n=2或(b) 2d=2又或(c) n=3且2d=4。之後，有人在射影多樣體上討論一樣的問題，發現這個實代數幾何上問題竟然和複代數幾何中一個很重要的物件極小多項式有緊密的關係。我們將用這項定理來處理一些已知的結果。

There are many guess whether a non-negative real polynomial can be written
as a sum-of-squares of polynomial. Until Hilbert proof that every non-negative
homogeneous polynomial of degree 2d in n-variables can be written as a sum-ofsquares of homogeneous polynomials of degree d if and only if either (a) n=2 or
(b) 2d=2 or (c) n=3 and 2d=4. After that, some people ask the same question
on the projective variety. They discover this problem which is the problem in the
real algebraic geometry has a closed relation with the variety of minimal degree
which is an important object in complex algebraic geometry. We will use the new
theorem to deal with some known results.

1 Introduction...6
2 Sum of sqrares...8
3 Variety of Minimal Degree...10
3.1 Invariants of projective variety...10
3.2 Property of degree and dimension...13
3.3 Variety of minimal degree...16
3.4 Classification of Variety of Minimal Degree...17
4 The Main Theorem of Sum-of-squares and Variety of Minimal
degree...19
4.1 sum of squares imply minimal degree...19
4.2 minimal degree imply sum of squares...21
5 Application...25
6 Conclusion...28
Bibliography...29

David Hilbert, Uber die Darstellung definiter Formen als Summe von For- ¨
menquadraten, Math. Ann. 32, no. 3, 342–350, 1888.
E. Artin, Uber die Zerlegung definiter Funktionen in Quadrate, Abh. math.
Sem. Hamburg 5, 110–115, 1927.
T. S. Motzkin, The arithmetic-geometric inequality. In: Proc. Symposium
on Inequalities, edited by O. Shisha, Academic Press, New York, 205–224,
1967.
M. D. Choi, T. Y. Lam, and B. Reznick. Sums of squares of real polynomials. In K-theory and algebraic geometry: connections with quadratic forms
and division algebras (Santa Barbara, CA, 1992), volume 58 of Proc. Sympos.
Pure Math., 103–126. Amer. Math. Soc., Providence, RI, 1995.
G. Blekherman, G. G. Smith, and M. Velasco. Sum of squares and varities
of minimal degree. J. Amer. Math. Soc., 29(3):893-913, 2016.
D. Eisenbud, J. Harris: On varieties of minimal degree (a centennial account),
in “Algebraic geometry, Bowdoin, 1985 (Brunswick, Maine, 1985)”, 3–13,
Proc. Sympos. Pure Math. 46, Part 1.
G. Blekherman, G. G. Smith, and M. Velasco. Sharp degree bounds for
sum-of-square certificates on algebraic curve. Journal de Mathmatiques Pures
et Appliques (JMPA), Volume 129, 61-86, 2019.
29
D. Mumford: Algebraic Geometry I: Complex Projective Varieties,
Springer, New York, 1976.
Igor R. Shafarevich: Basic Algebraic Geometry 1: Variety in Projective
Space Third Edition, Springer, New York, 2013.
R. Hertshorne: Algebraic Geometry, Springer, New York, 1977.
E. Bertini: Introduzione alla geometria proiettiva degli iperspazi, Enrico
Spoerri, Pisa, 1907.
P. Del Pezzo: Sulle superficie di ordine n immerse nello spazio di n+1 dimensioni, Rend. Circ. Mat. Palermo 1, 1886.
M. Velasco. Algebraic geometry through thr sum-of-squares lens.